Digital Signal Processing
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1 Digital Signal Proceing IIR Filter Deign Manar Mohaien Office: F8 School of IT Engineering
2 Review of the Precedent Lecture Propertie of FIR Filter Application of FIR Filter FIR Deign Method Windowing Method Frequency-ampling Method Leat-quare Method Application of FIR Filter Differentiator Hilbert Tranform
3 Cla Objective Introduce for IIR Filter Introduce the Zero-pole Placement Deign Method Introduce Claical Analog Filter Butterworth Filter Chebyhev-I Chebyhev-II Introduce Other IIR Filter deign Method Bilinear Tranformation Method Frequency Tranformation
4 IIR Filter Deign Method Introduction b + bz + L+ b m mz az L a n nz Hz () H(z) i an IIR filter if a i for ome i n. Otherwie, H(z) i an FIR filter.
5 Pole-zero Placement Ued for pecialized filter that arie in application. IIR Filter Deign by Pole-zero Placement Example Reonator Notch Filter Comb Filter
6 Reonator Narrowband filter Pae a ingle frequency Called a reonator with a reonant frequency F. The frequency repone i given by: IIR Filter Deign by Pole-zero Placement H ( f) δ ( f F ), f f / re δ ( f ), f, f The angle i correponding to the frequency f. The angle π i correponding to the frequency f f/. Therefore, π F θ f i the angle correponding to F.
7 Reonator contd. To obtain a table filter The (pole) radiu mut be le than. Alo, the reonator attenuate the end frequencie H() & H(-) i.e., zero at z and z -. IIR Filter Deign by Pole-zero Placement The aforementioned contraint yield Hz () b ( z )( z+ ) [ z rexp( jθ )][ z rexp( jθ )] b ( z ) co( θ ) + z r z r
8 IIR Filter Deign by Pole-zero Placement Hz () b ( z )( z+ ) [ z rexp( jθ )][ z rexp( jθ )] b ( z ) co( θ ) + z r z r Two parameter to be determined r (pole radiu) The bet i. It i approximated by ( F - ΔF - 3-dB frequency) r ΔFπ f b It i choen o that the paband gain i one. b exp( j θ ) rco( θ )exp( jθ ) + r exp( j θ )
9 IIR Filter Deign by Pole-zero Placement Reonator contd. Example 8. F Hz, deign a reonator to meet F Hz, Δ F 6Hz Solution b π F π () π 3 r 6π.9843 θ f exp( j θ ) rco( θ )exp( jθ ) + r.56 exp( j θ ) Im(z) Pole-zero plot H ) - - () z.56( z Re(z).9843z z re -
10 Reonator contd. Example 8. contd. IIR Filter Deign by Pole-zero Placement Magnitude repone.8.6 A(f) f (Hz)
11 Notch Filter A harp bandtop filter Therefore We place a zero at the angle of F. IIR Filter Deign by Pole-zero Placement H () z δ ( f F ), f f / notch H notch () z b [ z exp( jθ )][ z exp( jθ )] [ z rexp( jθ )][ z rexp( jθ )] The pole i inerted to control the 3-dB bandwidth of the topband. Finally, H b ( z co( θ ) z+ ) z co( r ) z+ () z notch θ b rco( θ ) + r co( θ )
12 Notch Filter Example 8. IIR Filter Deign by Pole-zero Placement.5.5 Pole-zero plot f 4 Hz, F 8Hz, Δ F 8Hz π F π (8) π 4 3 θ f r 8π Im(z) Re(z) b rco( θ ) + r.9766 co( θ ).4. Magnitude repone H () z.9766( + z + z ) notch z z A(f) f (Hz)
13 Comb Filter IIR Filter Deign by Pole-zero Placement comb floor( n/) H ( f) δ ( f if ), f f / i with F f / n, with n denote the filter order. comb () b H z n n, b r rz n Invere Comb Filter comb floor( n/) H ( f) δ ( f if ), f f / i b ( z n) H () z, r inv n n b rz + n
14 Comb Filter IIR Filter Deign by Pole-zero Placement Pole-zero plot Pole-zero plot Im(z) Im(z) Re(z) Re(z) Comb filter (n ) Invere Comb filter (n )
15 IIR Filter Deign by Pole-zero Placement Magnitude repone.8 A(f) f (Hz) Magnitude repone.8 A(f) f (Hz)
16 IIR Filter Deign Parameter Deign Method Start with a normalized analog filter called prototype filter. Paband -δ p A(f) δ A ( f), f F p a p A ( f) δ, F f a δ F p F f / Paband ripple and topband attenuation in db are given by A f (Hz) Stopband log ( δ )db, A log ( δ )db p p
17 IIR Filter Deign Parameter contd. Selectivity Factor r F F p < r <, with r for an ideal filter. Dicrimination Factor d / ( ) δ p δ d > for practical filter. d for an ideal filter.
18 IIR Filter Deign Parameter contd. Analog Filter We tart with a deired magnitude repone and working backward to obtain the pole, zero, and gain. H ( f) H ( ) a a jπ f A ( f) H ( f) a H a a () jπ f H H * a() a() jπ f H () H ( ) a a jπ f H H A f () ( ) a a a( ) f /( j π )
19 Claical Analog Filter Butterworth Filter A Butterworth filter of order n i a LP filter with the following A a ( f) + ( f / F ) n Aa ( F c ).5 Frequency Fc i called the 3-dB cutoff frequency log ( ) 3dB { Aa Fc } Maximally flat paband filter. c (-δ p ).8 Fc Hz, n 4. A a (f).6.4. δ.5 F p F f (Hz)
20 Claical Analog Filter contd. Butterworth Filter contd. H H A f () ( ) a a a( ) f /( j π ) + /( j π F ) c n p π F exp( jθ ), k < n k c k ( j π F ) ( j π F ) + c n c n n with θ (k + + n) π k n ( ) ( π F ) ( ) ( ) + n n c n π F n n c For normalized lowpa filter Fc /π Hz (equivalent to rad/ec)
21 Claical Analog Filter contd. Butterworth Filter contd. Odd order: n 5 Even order: n 6 Im() Im() Re() Re() H a () ( π F ) n c ( p )( p)...( p ) n
22 Claical Analog Filter contd. Propertie of the Butterworth Filter The magnitude repone decreae monotonically. n db per decade. { a } { a } log A ( f) log A ( f) n The firt (n ) derivative of A (f) equal at f. The magnitude repone of the Butterworth filter i a flat a poible at f. [Butterworth Filter are maximally flat filter] Filter order + ( Fp/ Fc) + ( F / F ) c n n ( δ ) ( δ ) p ceil n ln( d) ln( r ) The paband and topband pecification are exceeded.
23 Claical Analog Filter contd. Filter order contd. To meet exactly the contraint + ( Fp/ Fc) + ( F / F ) c n n ( δ ) ( δ ) p F cp F c F p /( n) ( ) δ p δ F /( n) F c F cp + F c
24 Claical Analog Filter contd. Butterworth Filter Deign a LP filter with the following pec. Solution F Hz, F Hz, δ.5, δ.5 p p r Fp.5 F d / / ( δ ) (.95) p δ (.5).65 ceil ln(.65) n 6 ln(.5) F cp F 3.8 Hz p /( n) ( ) δ p F c F 9. Hz δ /( n) F c Fcp + Fc 6. Hz
25 Claical Analog Filter contd. Chebyhev-I Filter Butterworth Filter Smooth paband and mooth tranition band. Wide tranition band. Chebyhev-I It allow ripple in the paband. It reduce the width of the tranition band. A a ( f) + ε T f F n( / p) Where n i filter order and ε i a ripple factor >. T n (x) i a plynomial of degree n called a Chebyhev polynomial.
26 Claical Analog Filter contd. Chebyhev-I Filter contd. Chebyhev Polynomial Odd function for odd n and even function for even n. T n () for all n. T n (x) ocillate in [-, ] for x and monotonic when x >. Thee polynomial lead to equal ripple in the paband. n Tn(x) x 3 x - 4 4x 3 3x T () x xt () x + T (), x k k+ k k
27 Claical Analog Filter contd. Chebyhev-I Filter contd. Equiripple paband filter /(+ε ).8 A a (f).6.4. δ.5 F f (Hz) a A A ( F ) p ( δ ) ( ) / + ε ( p ε δ p + ε ) a, n odd (), n even + ε
28 Claical Analog Filter contd. Chebyhev-I The pole are on an ellipe. The minor and major axe of the ellipe are α ε ε r ( ) + + πf α/ n α / n πf α/ n + α / n r ( ) The angle are given a (ame a thoe of the Butterworth filter) θ (k + + n) π, k < n k n The pole can be given in rectangular form p σ + jω k k k r co( θ ) + jr in( θ ), k < n k k
29 Claical Analog Filter contd. Chebyhev-I Filter Tranfer Function H () a(), ( ) n a β A β p p p ( p )( p) ( p ) L L n n ln( d + d ) r r + n ceil ln( )
30 Claical Analog Filter contd. Chebyhev-II Filter Smooth in the paband It ha ripple in the topband. The quare magnitude repone i given by: A a ( f) ε T ( F/ f) + ε T ( F / f) n n. Equiripple topband filter ( - δ p ).8 A a (f).6.4. ε /(+ε ).5 F p.5 f (Hz)
31 Claical Analog Filter contd. Chebyhev-II Filter contd. Propertie more A a( F) ε + ε ε δ( δ ) / a f ε ε + lim A ( ) f, n odd, n even Pole k ( π F ) p, k q k < n With p k a the pole of the Chebyhev-I filter. Chybehev-II ha n or n- finite zero for even and or odd n. Located along the imaginary axi at r jπ F, k in( k < n θ ) k
32 Claical Analog Filter contd. Chebyhev-II Filter contd. The DC gain of the quared magnitude repone i unity. H a () β( r )( r) L ( r ) n ( q )( q) L( q ) n β qqlq rrlr n n
33 Claical Analog Filter contd. Elliptic Filter They have equal ripple at both topband and paband. Achieve narrow tranition band. A a ( f) + ε U f F n( / ) U n i an nth order Jacobian elliptic function Chebyhev rational function. Elliptic filter are more complex than Butterworth or Chebyhev filter. Finding the pole and zero require iterative olution of nonlinear algebraic equation.
34 Claical Analog Filter contd. Elliptic Filter contd. To find the filter order, conider the following function gx () π / dθ x in ( θ) Then the filter order i given by: n ceil ( ) ( ) ( ) ( ) gr g d gd g r
35 Bilinear-tranformation Method Starting From the Analog Filter H a () b () ( p )( p) L( p ) n Replace the integrator (/) with an approximation of the form yk ( ) yk ( ) + T xk ( ) + xk ( ) H T + z () z z gz () / H() z z T z + Finally, Hz () H() a gz ()
36 Bilinear Mapping Bilinear-tranformation Method contd. More detail, Hz () H() a gz () z + T ( + σt) + ( ωt) z T ( σt) + ( ωt) σ + j The bilinear mapping map the imaginary part of the plane into the unit circle of the z tranform. ω If σ <, then z < Left half of the plane mapped into inide the unit circle of the z plane. If σ >, then z > Right half of the plane i mapped outide the unit circle of z lane.
37 Graphical Repreentation of the Bilinear Mapping Bilinear-tranformation Method contd. Analog plane Digital z plane πt Im() Im(z) Re() Re(z)
38 Bilinear-tranformation Method contd. Remark The analog Frequency range F < i mapped to f < f/. jπf, therefore and j F exp( j π ft) π, T exp( j π ft) + jin( π ft) jtan( π ft) T co( π ft) T exp( jπ ft) exp( jπ ft) exp( jπ ft) T exp( jπ ft) exp( jπ ft) + exp( jπ ft) F tan( π ft) πt
39 Mapping From continuou frequency to digital frequency Bilinear-tranformation Method contd. F tan( π ft) tan ( π ft) f πt πt 6 Freqeuncy warping 5 4 F (Hz) f/f
40 Analog Frequency Tranformation Analog Frequency Tranformation Start from the normalized lowpa Butterworth filter Hnorm () + H a () Lowpa with cutoff Ω Highpa with cutoff Ω Bandpa with cutoff Ω, Ω Bandtop with cutoff Ω, Ω Ω Ω + ΩΩ ( Ω Ω) ( Ω Ω ) + ΩΩ
41 How to Deign a Digital Filter Map value into analog domain Obtain the normalized lowpa filter Apply analog frequency tranformation Conider the cutoff frequencie and filter type. Apply bilinear tranformation Replace with g(z). Analog Frequency Tranformation contd. analog analog digital analog frequency tranformation normalized lowpa filter bilinear tranformation
42 Example 8. Deign a bandpa digital filter with f 5 Hz, f 5 Hz, and f 5 Hz. Analog Frequency Tranformation contd. STEP : Analog frequency mapping Map frequencie into continuou frequency domain Analog normalized lowpa filter STEP : Analog frequency tranformation tan( π ft) tan( π ft Ω ) Ω π πt π 3.49 πt H () H () a norm Hnorm +ΩΩ ( Ω Ω ) () + ( Ω Ω ) ( Ω Ω ) + Ω Ω
43 Analog Frequency Tranformation contd. Example 8. contd. STEP 3: Bilinear tranformation.48( z ) Hz () Ha[ gz ()].445z +.584z gz () z T z + Magnitude repone.8 A(f) f (Hz)
44 Digital Tranformation Method Deign method analog digital digital normalized lowpa filter bilinear tranformation digital frequency tranformation
45 Summary and Dicuion Introduced for IIR Filter Introduced the Zero-pole Placement Deign Method Introduced Claical Analog Filter Butterworth Filter Chebyhev-I Chebyhev-II Introduced Other IIR Filter deign Method Bilinear Tranformation Method Frequency Tranformation
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